I think you are talking about the Oatey “Twist-N-Set” replacement flange. This flange has a neoprene gasket that screws onto the threaded section of the replacement flange and is meant to be compression-fitted into the 4″ closet bend. You can insert this flange on top of the old flange, however it will then sit atop the old flange and the toilet will not seat properly.
I have indeed done just that in a few cases where the old flange was set too low in the first place, either because it was improperly installed or because the floor was built up. I don’t know how many toilets I’ve replaced where someone has tiled a bathroom floor and the toilet was set on a flange now an inch below the surface! Most of them didn’t leak that bad, a testament to the ‘forgiveness” of wax seals, especially those with ample plastic sleeves.
If your old flange is at the proper level, however, you’ll need to remove it, as retired plbg says.
You might wonder whether these inserted flanges restrict toilet outflow, since they are fitted inside the 4″ closet bend. This really isn’t a problem, since the toilet trap discharge opening is still less-than-or equal-to the opening in the Twist-N-Set.
Hope this helps.